Data can be:
Data can also be classified as
A variable is a name given to a set of data, e.g. Let X be the heights of 100 selected people. A variable is denoted by a (capital) letter.
A particular value of X is called a score, e.g the height of the tenth person is 74.44cm. To talk about scores in general, use the lower case version of the variable. So x is a score relating to the variable X.
Large amounts of data often need to be summarised in order to make it easier to analyse.
Stem and leaf plots are a method for showing the frequency with which certain classes of values occur. A stem and leaf plot is a two column table - stem and leaf. Each row in the table represents data points with the same stem but different leafs.
There is a lot of choice in how you decide on which part of a number is a stem and which part is a leaf.
For two digit numbers, you can make the stem the left-most digit of the number. The leaf is the final digit of the number. The digits in the leaf are arranged in increasing order. One leaf for each data item, so repeated leaf values can occur.
For numbers with more than two digits, other types of stem/leaf division may need to be used.
Example Here are the scores of 13 students in a test:
14 | 21 | 32 | 43 | 52 | 65 | 73 | 16 | 25 | 39 | 44 | 57 | 28 |
and here is the stem and leaf plot
When summarising a set of data, making a stem and leaf plot before you do anything else helps to get a feel for the data.
Example Here are the lengths in centimetres of 45 examples of a certain species of tropical fish:
14 | 19 | 15 | 30 | 25 | 24 | 36 | 21 | 43 |
20 | 13 | 18 | 16 | 31 | 26 | 23 | 37 | 20 |
27 | 21 | 12 | 17 | 17 | 32 | 27 | 22 | 38 |
22 | 26 | 22 | 11 | 16 | 18 | 33 | 28 | 21 |
41 | 23 | 28 | 23 | 32 | 15 | 19 | 34 | 29 |
and a possible stem and leaf plot is
You can see from the stem and leaf plot that just using the first digit to break the data values into groups may not give a very good picture of the spread of the data. So the grouping may need to be refined.
Guided Examples
Data can be plotted with Dot plots – vertical dots to represent data values
Example Here are the brands of mobile phones owned by 26 people
Brand | Number |
Apple | 5 |
Samsung | 14 |
Sony | 4 |
Motorola | 2 |
Telstra | 1 |
and here is a dot plot of the data:
A frequency table summarises the data by grouping it in ranges of values. The ranges are called classes.
A frequency table has at least two columns - one for the classes the data is grouped into and one for the number of data items in each class. This column is the frequency.
The process for constructing a frequency table is
There are some terms associated with the resulting table.
Example Continuing with our sample of 45 examples of a certain species of tropical fish, you saw from the stem and leaf pot that just using four classes did not give a very enlightening picture of the spread of the data. Using more classes will help.
For this data, seven classes was selected. To calculate the class width you need to find the maximum and minimum score in the data. Then to calculate the class width: \[ \text{max - min} = 43 - 11 = 32 \] and if you round this up to 35 to make it divisble by 7, this gives a class width of 5.
Here is the frequency table constructed from the data:
Once you have a frequency table you can use it to create different visual presentations of your data.